Exploring Mandelbrot Fractals
On the left you see the Mandelbrot set.
To calculate the following formular is used:
z_{n+1} = z_{n}^{2} + c
This means that we build element after element (z_{n+1}) of a chain depending on the previous element (z_{n}) by squaring the previous and adding a constant element c.
Now we run the formular mapping each pixel of the canvas to a value of c = x + yi.
Starting with z_{0} = 0 this chain may explode (|z_{n}| > 2) for some values of c after some time n.
- If it explodes the coordinate for c is colored depending on n
- if not the point c is inside the mandelbrot-set and it is colored dark gray.
When clicking into the picture the value of c (the coordinate) is shown and the path of z_{n} is drawn in black.
Control with Mouse:
- Ctrl + Left mouse button hold -> Zoom-Area
- Shift + Left mouse button hold -> Move-Area
- Ctrl + Right mouse button click -> Reset Zoom
Mandelbrot Set
Julia set
If we take the same formular as above (z_{n+1} = z_{n}^{2} + c) and instead of mapping each pixel of the canvas to a value of c we map it to the starting z_{0}=x+yi we get a slightly different fractal (click into the mandelbrot picture for setting c):